Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
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Chapter 7.6, Problem 6E
In Problems
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Suppose solving an equation by Laplace transform results in
4 (s+ 18)
Y(s)
(s +9)² *
Evaluate y(0).
Problem 1
Determine whether each of the following functions T : R² → R² is linear. Prove your answer.
a. T(a₁, a2) = (1, a2)
b. T(a₁, a2) = (a₁ + a2, 2a1 − a2)
c. T(a₁, a2) = (a₁ + 1, a2)
In Problems 1-12, use Definition 1 to determine the Laplace
transform
of the given function.
Chapter 7 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1-12, use Definition 1 to determine...Ch. 7.2 - In Problems 1 -12, Use Definition 1 to determine...
Ch. 7.2 - In Problems 112, use Definition 1 to determine the...Ch. 7.2 - In Problems 112, use Definition 1 to determine the...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - Prob. 14ECh. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 13-20, use the Laplace transform table...Ch. 7.2 - In Problems 2128, determine whether f(t) is...Ch. 7.2 - Prob. 22ECh. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - In Problems 21-28, determine whether f(t) is...Ch. 7.2 - Which of the following functions are of...Ch. 7.2 - For the transforms F(s) in Table 7.1, what can be...Ch. 7.2 - Thanks to Eulers formula page 166 and the...Ch. 7.2 - Prob. 32ECh. 7.2 - Prove that if f is piecewise continuous on [a,b]...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - Prob. 6ECh. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - Prob. 9ECh. 7.3 - In Problems 1-20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - In Problems 1- 20, determine the Laplace transform...Ch. 7.3 - Prob. 19ECh. 7.3 - Prob. 20ECh. 7.3 - Given that L{cosbt}(s)=s/(s2+b2), use the...Ch. 7.3 - Starting with the transform L{1}(s)=1/s, use...Ch. 7.3 - Use Theorem 4 on page 362 to show how entry 32...Ch. 7.3 - Show that L{eattn}(s)=n!/(sa)n+1 in two ways: a....Ch. 7.3 - Use formula (6) to help determine. a. L{tcosbt}....Ch. 7.3 - Prob. 26ECh. 7.3 - Prob. 27ECh. 7.3 - Prob. 28ECh. 7.3 - The transfer function of a linear system is...Ch. 7.3 - Prob. 30ECh. 7.3 - Prob. 31ECh. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Prob. 34ECh. 7.3 - Prob. 35ECh. 7.3 - Prob. 36ECh. 7.3 - Initial value theorem. Apply the relation...Ch. 7.3 - Prob. 38ECh. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - Prob. 5ECh. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - In Problems 1-10, determine the inverse Laplace...Ch. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - Prob. 14ECh. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - In Problems 11-20, determine the partial fraction...Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - Prob. 22ECh. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - In Problems 21-30, determine L1{F}....Ch. 7.4 - Determine the Laplace transform of each of the...Ch. 7.4 - Prob. 32ECh. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Theorem 6 in Section 7.3 on page 364 can be...Ch. 7.4 - Prob. 37ECh. 7.4 - Prob. 38ECh. 7.4 - Prob. 39ECh. 7.4 - Heavisides Expansion Formula. Let P(s) and Q(s) be...Ch. 7.4 - Prob. 41ECh. 7.4 - Prob. 42ECh. 7.4 - Prob. 43ECh. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 114, solve the given initial value...Ch. 7.5 - In Problems 1-14, solve the given initial value...Ch. 7.5 - In Problems 114, solve the given initial value...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 1524, solve for Y(s), the Laplace...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2528, solve the given third-order...Ch. 7.5 - In Problems 2932, use the method of Laplace...Ch. 7.5 - In Problems 2932, use the method of Laplace...Ch. 7.5 - In Problems 29-32, use the method of Laplace...Ch. 7.5 - In Problems 29-32, use the method of Laplace...Ch. 7.5 - Prob. 33ECh. 7.5 - Use Theorem 6 in Section 7.3, page 364, to show...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - In Problems 3538, find solutions to given initial...Ch. 7.5 - Determine the error e(t) for the automatic pilot...Ch. 7.5 - Prob. 40ECh. 7.5 - Prob. 41ECh. 7.6 - In Problems 14, sketch the graph of the given...Ch. 7.6 - Prob. 2ECh. 7.6 - In Problems 14, sketch the graph of the given...Ch. 7.6 - Prob. 4ECh. 7.6 - In Problems 510, express the given function using...Ch. 7.6 - In Problems 510, express the given function using...Ch. 7.6 - Prob. 7ECh. 7.6 - In Problems 5-10, express the given function using...Ch. 7.6 - Prob. 9ECh. 7.6 - Prob. 10ECh. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - Prob. 12ECh. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - Prob. 16ECh. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - In Problems 1118, determine an inverse Laplace...Ch. 7.6 - The current I(t) in an RLC series circuit is...Ch. 7.6 - The current I(t) in an LC series circuit is...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 2124, solve the given initial value...Ch. 7.6 - In Problems 25-32, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 25-32, solve the given initial value...Ch. 7.6 - In Problems 2532, solve the given initial value...Ch. 7.6 - In Problems 25-32, solve the given initial value...Ch. 7.6 - Prob. 35ECh. 7.7 - In Problems 1-4, determine L{f}, where f(t) is...Ch. 7.7 - Prob. 2ECh. 7.7 - Prob. 3ECh. 7.7 - In Problems 1-4, determine L{f}, where f(t) is...Ch. 7.7 - Prob. 5ECh. 7.7 - Prob. 6ECh. 7.7 - In Problems 5-8, determine L{f}, where the...Ch. 7.7 - Prob. 8ECh. 7.7 - Show that if L{g}(s)=[(s+)(1eTs)]1, where T0 is...Ch. 7.7 - Prob. 10ECh. 7.7 - Prob. 11ECh. 7.7 - Prob. 12ECh. 7.7 - Prob. 14ECh. 7.7 - Prob. 15ECh. 7.7 - Prob. 16ECh. 7.7 - In Problems 1518, find a Taylor series for f(t)...Ch. 7.7 - Prob. 18ECh. 7.7 - Prob. 19ECh. 7.7 - Use the recursive relation (7) and the fact that...Ch. 7.7 - Prob. 21ECh. 7.7 - Prob. 22ECh. 7.7 - Prob. 23ECh. 7.7 - Use the procedure discussed inProblem 23 to show...Ch. 7.7 - Find an expansion for ln[1+(1/s2)] in powers of...Ch. 7.7 - Prob. 26ECh. 7.7 - Prob. 27ECh. 7.8 - In Problems 14, use the convolution theorem to...Ch. 7.8 - Prob. 2ECh. 7.8 - Prob. 3ECh. 7.8 - Prob. 4ECh. 7.8 - Prob. 5ECh. 7.8 - Prob. 6ECh. 7.8 - Prob. 7ECh. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - Prob. 9ECh. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - In Problems 512, use the convolution theorem to...Ch. 7.8 - Prob. 13ECh. 7.8 - Find the Laplace transform of f(t):=0tevsin(tv)dvCh. 7.8 - Prob. 15ECh. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - Prob. 17ECh. 7.8 - Prob. 18ECh. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - In Problems 1522, solve the given integral...Ch. 7.8 - Prob. 22ECh. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - Prob. 24ECh. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - In Problems 2328, a linear system is governed by...Ch. 7.8 - Prob. 29ECh. 7.8 - In Problems 29 and 30, the current I(t) in an RLC...Ch. 7.8 - Prob. 31ECh. 7.8 - Prob. 32ECh. 7.8 - Prob. 33ECh. 7.8 - Prob. 34ECh. 7.8 - Prob. 35ECh. 7.8 - Prob. 36ECh. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - Prob. 2ECh. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - Prob. 4ECh. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - In Problems 1-6, evaluate the given integral....Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - Prob. 11ECh. 7.9 - In Problems 7-12, determine the Laplace transform...Ch. 7.9 - Prob. 13ECh. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - Prob. 15ECh. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - Prob. 19ECh. 7.9 - In Problems 13-20, solve the given symbolic...Ch. 7.9 - In Problems 21-24, solve the given symbolic...Ch. 7.9 - Prob. 22ECh. 7.9 - In Problems 21-24, solve the given symbolic...Ch. 7.9 - Prob. 24ECh. 7.9 - Prob. 25ECh. 7.9 - Prob. 26ECh. 7.9 - Prob. 27ECh. 7.9 - Prob. 28ECh. 7.9 - Prob. 29ECh. 7.9 - Prob. 30ECh. 7.9 - A linear system is said to be stable if its...Ch. 7.9 - A linear system is said to be asymptotically...Ch. 7.9 - Prob. 33ECh. 7.9 - Prob. 34ECh. 7.9 - Figure 7.29 shows a beam of length 2 that is...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - Prob. 9ECh. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - Prob. 17ECh. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - In Problems 1-19, use the method of Laplace...Ch. 7.10 - Use the method of Laplace transforms to solve...Ch. 7.10 - Recompute the coupled mass-spring oscillator...Ch. 7.10 - In Problems 23 and 24, find a system of...Ch. 7.10 - In Problems 23 and 24, find a system of...Ch. 7.RP - In Problems 1 and 2, use the definition of the...Ch. 7.RP - In Problems 1 and 2, use the definition of the...Ch. 7.RP - Prob. 3RPCh. 7.RP - In Problems 3-10, determine the Laplace transform...Ch. 7.RP - In Problems 3-10, determine the Laplace transform...Ch. 7.RP - In Problems 3-10, determine the Laplace transform...Ch. 7.RP - Prob. 7RPCh. 7.RP - Prob. 8RPCh. 7.RP - Prob. 9RPCh. 7.RP - Prob. 10RPCh. 7.RP - Prob. 11RPCh. 7.RP - In Problems 11-17, determine the inverse Laplace...Ch. 7.RP - Prob. 13RPCh. 7.RP - Prob. 14RPCh. 7.RP - Prob. 15RPCh. 7.RP - Prob. 16RPCh. 7.RP - Prob. 17RPCh. 7.RP - Prob. 18RPCh. 7.RP - Prob. 19RPCh. 7.RP - In Problems 19-24, solve the given initial value...Ch. 7.RP - Prob. 21RPCh. 7.RP - Prob. 22RPCh. 7.RP - Prob. 23RPCh. 7.RP - Prob. 24RPCh. 7.RP - In Problems 25 and 26, find solutions to the given...Ch. 7.RP - In Problems 25 and 26, find solutions to the given...Ch. 7.RP - Prob. 27RPCh. 7.RP - Prob. 28RPCh. 7.RP - A linear system is governed by y5y+6y=g(t). Find...Ch. 7.RP - Prob. 30RPCh. 7.RP - Prob. 31RPCh. 7.RP - In Problems 31 and 32, use Laplace transforms to...
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- A. Find the Laplace Transform if the given is a function of t, and Inverse Transform if the given is a function of s. s+2 3. F(s)= s² – 3s +4arrow_forwardExpress the graph in Figure Q3(a) in terms of a unit step function and then find theLaplace transform for the function.arrow_forwardIf * represents convolution, determine the Laplace transform of f(t) = 12t *e-3tarrow_forward
- 7. Find the inverse Laplace transform of the following function S-3 4s²-3s+5 (a) (s2-6s+10) (b) (s+1)(s²-35+2) e-3s (c) s2+4s+13 (d) (s²+1)²arrow_forward4. (a) Show that T1 and T2 are equivalent statistics if, and only if, we can write T2 = H(T1) for some 1-1 transformation H of the range of T into the range of T2. Which of the following statistics are equivalent? (Prove or disprove.) (b) II, *, and log x,, c; >0 (c) -1 Ti and E-, log ri, x; > 0 (d) (Σ" σ,, ΣΕ ο?) and (Σ o ΣL (οι- 3)?) (e) (E, Ti, E-1 ) and (E ri, E (*; – 1)*). Xi,arrow_forward6. Find the inverse Laplace transform of the function I(s) = (3² − a²) (as + b) (s² + a²) (s² + ß²) *arrow_forward
- when using the Laplace Transform Table, you need to always identify c, f(t), and F(s), etc. 2 Find L-1 3s -arrow_forwardWrite down the qualitative form of the inverse Laplace transform of the followir functions. For each question first write down the poles of the function, X(s). (a) s+1 X(s) = (s+2)(s² + 2s + 2)(s² + 4) (b) 1 X(s) = %3D (2s2 + 8s + 20)(s² + 2s + 2)(s+ 8) (c) s²(s² + 2s + 5)(s+3)arrow_forwardQuestion 1 + 00 The definition of the Laplace transform of a function is that given a piecewise defined function, L{f(t) } = / e"f(1) dt, if it exists. (A) True B) Falsearrow_forward
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